Properties of Dirichlet Convolution

Theorem

Let $f, g, h$ be arithmetic functions.

Let $*$ denote Dirichlet convolution.

Let $\iota$ be the identity arithmetic function.


Then the following properties hold:


Dirichlet Convolution is Commutative

$f * g = g * f$


Dirichlet Convolution is Associative

$\paren {f * g} * h = f * \paren {g * h}$


Identity Element for Dirichlet Convolution

$\iota * f = f$


Dirichlet Convolution Preserves Multiplicativity

Let $f, g: \N \to \C$ be multiplicative arithmetic functions.


Then their Dirichlet convolution $f * g$ is again multiplicative.


Also see

  • Definition:Ring of Arithmetic Functions
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